Composite and Inverse Functions

Description

At A Glance

• The composition of functions is defined by using the output of one function as the input of another.
• To evaluate the composition of two functions $f(g(x))$ for a value of $x$, first find $g(x)$ and then substitute the value in $f$, or find a general rule for $f(g(x))$ and then evaluate for the value of $x$.
• To determine the rule for a composite function $f(g(x))$, substitute $g(x)$ into $f(x)$ and simplify. The domain of $f(g(x))$ is the set of values in the domain of $g$ for which $g(x)$ is in the domain of $f$.
• Composite functions can be decomposed into two or more functions using methods such as the properties of operations, the properties of exponents, and the word form of the function.
• Since composition of functions is associative, the way in which three or more functions are grouped in a composite function does not change the result.
• Composition is not commutative, so composing functions in a different order does not necessarily yield the same result.
• The identity function for the operation of composition is $f(x) = x$.
• A function is one-to-one if every output value has a unique input value. The horizontal line test and algebraic methods are used to determine whether a function is one-to-one.
• Two functions $f$ and $g$ are inverse functions if $f(g(x)) = g(f(x)) = x$. To determine the rule for an inverse function, write as $y = f(x)$, reverse $x$ and $y$, and then solve for $y$. The domain of $f$ is the range of $g$, and vice versa.
• The graphs of inverse functions are reflections of each other across the line $y = x$.
• If the inverse of a function is not a function, the domain of the function can be restricted to values that produce a function for the inverse.